Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [75,10,Mod(1,75)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("75.1"); S:= CuspForms(chi, 10); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(75, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 10, names="a")
 
Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 75.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(38.6276877123\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 1546x^{2} + 152x + 559560 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3}\cdot 3\cdot 5^{2}\cdot 23 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-27.5964\) of defining polynomial
Character \(\chi\) \(=\) 75.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-38.7320 q^{2} -81.0000 q^{3} +988.165 q^{4} +3137.29 q^{6} -6373.44 q^{7} -18442.8 q^{8} +6561.00 q^{9} +67637.2 q^{11} -80041.4 q^{12} +162029. q^{13} +246856. q^{14} +208386. q^{16} -308578. q^{17} -254120. q^{18} -526216. q^{19} +516249. q^{21} -2.61972e6 q^{22} -1.35822e6 q^{23} +1.49387e6 q^{24} -6.27569e6 q^{26} -531441. q^{27} -6.29801e6 q^{28} +6.02016e6 q^{29} -7.89958e6 q^{31} +1.37153e6 q^{32} -5.47861e6 q^{33} +1.19518e7 q^{34} +6.48335e6 q^{36} -6.52067e6 q^{37} +2.03814e7 q^{38} -1.31243e7 q^{39} -3.65494e6 q^{41} -1.99953e7 q^{42} -2.63401e7 q^{43} +6.68367e7 q^{44} +5.26067e7 q^{46} +1.04707e7 q^{47} -1.68792e7 q^{48} +267157. q^{49} +2.49948e7 q^{51} +1.60111e8 q^{52} +1.10225e8 q^{53} +2.05838e7 q^{54} +1.17544e8 q^{56} +4.26235e7 q^{57} -2.33173e8 q^{58} +2.16362e6 q^{59} -1.44173e8 q^{61} +3.05966e8 q^{62} -4.18162e7 q^{63} -1.59816e8 q^{64} +2.12197e8 q^{66} +1.09836e8 q^{67} -3.04926e8 q^{68} +1.10016e8 q^{69} +1.72652e8 q^{71} -1.21003e8 q^{72} +2.71753e8 q^{73} +2.52559e8 q^{74} -5.19988e8 q^{76} -4.31082e8 q^{77} +5.08331e8 q^{78} +6.32088e8 q^{79} +4.30467e7 q^{81} +1.41563e8 q^{82} -3.09787e7 q^{83} +5.10139e8 q^{84} +1.02020e9 q^{86} -4.87633e8 q^{87} -1.24742e9 q^{88} -3.96023e7 q^{89} -1.03268e9 q^{91} -1.34215e9 q^{92} +6.39866e8 q^{93} -4.05552e8 q^{94} -1.11094e8 q^{96} +9.30586e8 q^{97} -1.03475e7 q^{98} +4.43767e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} - 324 q^{3} + 1792 q^{4} - 162 q^{6} - 13036 q^{7} + 24636 q^{8} + 26244 q^{9} + 104696 q^{11} - 145152 q^{12} - 140812 q^{13} - 181062 q^{14} + 1319800 q^{16} - 489352 q^{17} + 13122 q^{18}+ \cdots + 686910456 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −38.7320 −1.71173 −0.855864 − 0.517202i \(-0.826974\pi\)
−0.855864 + 0.517202i \(0.826974\pi\)
\(3\) −81.0000 −0.577350
\(4\) 988.165 1.93001
\(5\) 0 0
\(6\) 3137.29 0.988266
\(7\) −6373.44 −1.00330 −0.501652 − 0.865069i \(-0.667274\pi\)
−0.501652 + 0.865069i \(0.667274\pi\)
\(8\) −18442.8 −1.59192
\(9\) 6561.00 0.333333
\(10\) 0 0
\(11\) 67637.2 1.39289 0.696447 − 0.717608i \(-0.254763\pi\)
0.696447 + 0.717608i \(0.254763\pi\)
\(12\) −80041.4 −1.11429
\(13\) 162029. 1.57343 0.786715 − 0.617317i \(-0.211780\pi\)
0.786715 + 0.617317i \(0.211780\pi\)
\(14\) 246856. 1.71738
\(15\) 0 0
\(16\) 208386. 0.794928
\(17\) −308578. −0.896077 −0.448038 − 0.894014i \(-0.647877\pi\)
−0.448038 + 0.894014i \(0.647877\pi\)
\(18\) −254120. −0.570576
\(19\) −526216. −0.926345 −0.463173 − 0.886268i \(-0.653289\pi\)
−0.463173 + 0.886268i \(0.653289\pi\)
\(20\) 0 0
\(21\) 516249. 0.579258
\(22\) −2.61972e6 −2.38426
\(23\) −1.35822e6 −1.01204 −0.506018 − 0.862523i \(-0.668883\pi\)
−0.506018 + 0.862523i \(0.668883\pi\)
\(24\) 1.49387e6 0.919097
\(25\) 0 0
\(26\) −6.27569e6 −2.69328
\(27\) −531441. −0.192450
\(28\) −6.29801e6 −1.93639
\(29\) 6.02016e6 1.58058 0.790291 − 0.612732i \(-0.209929\pi\)
0.790291 + 0.612732i \(0.209929\pi\)
\(30\) 0 0
\(31\) −7.89958e6 −1.53630 −0.768150 − 0.640270i \(-0.778823\pi\)
−0.768150 + 0.640270i \(0.778823\pi\)
\(32\) 1.37153e6 0.231223
\(33\) −5.47861e6 −0.804188
\(34\) 1.19518e7 1.53384
\(35\) 0 0
\(36\) 6.48335e6 0.643337
\(37\) −6.52067e6 −0.571985 −0.285993 − 0.958232i \(-0.592323\pi\)
−0.285993 + 0.958232i \(0.592323\pi\)
\(38\) 2.03814e7 1.58565
\(39\) −1.31243e7 −0.908420
\(40\) 0 0
\(41\) −3.65494e6 −0.202001 −0.101000 − 0.994886i \(-0.532204\pi\)
−0.101000 + 0.994886i \(0.532204\pi\)
\(42\) −1.99953e7 −0.991532
\(43\) −2.63401e7 −1.17492 −0.587461 − 0.809252i \(-0.699872\pi\)
−0.587461 + 0.809252i \(0.699872\pi\)
\(44\) 6.68367e7 2.68830
\(45\) 0 0
\(46\) 5.26067e7 1.73233
\(47\) 1.04707e7 0.312994 0.156497 − 0.987678i \(-0.449980\pi\)
0.156497 + 0.987678i \(0.449980\pi\)
\(48\) −1.68792e7 −0.458952
\(49\) 267157. 0.00662039
\(50\) 0 0
\(51\) 2.49948e7 0.517350
\(52\) 1.60111e8 3.03673
\(53\) 1.10225e8 1.91884 0.959422 − 0.281973i \(-0.0909888\pi\)
0.959422 + 0.281973i \(0.0909888\pi\)
\(54\) 2.05838e7 0.329422
\(55\) 0 0
\(56\) 1.17544e8 1.59718
\(57\) 4.26235e7 0.534826
\(58\) −2.33173e8 −2.70552
\(59\) 2.16362e6 0.0232460 0.0116230 − 0.999932i \(-0.496300\pi\)
0.0116230 + 0.999932i \(0.496300\pi\)
\(60\) 0 0
\(61\) −1.44173e8 −1.33321 −0.666607 − 0.745409i \(-0.732254\pi\)
−0.666607 + 0.745409i \(0.732254\pi\)
\(62\) 3.05966e8 2.62973
\(63\) −4.18162e7 −0.334435
\(64\) −1.59816e8 −1.19072
\(65\) 0 0
\(66\) 2.12197e8 1.37655
\(67\) 1.09836e8 0.665896 0.332948 − 0.942945i \(-0.391957\pi\)
0.332948 + 0.942945i \(0.391957\pi\)
\(68\) −3.04926e8 −1.72944
\(69\) 1.10016e8 0.584299
\(70\) 0 0
\(71\) 1.72652e8 0.806323 0.403162 − 0.915129i \(-0.367911\pi\)
0.403162 + 0.915129i \(0.367911\pi\)
\(72\) −1.21003e8 −0.530641
\(73\) 2.71753e8 1.12001 0.560005 − 0.828489i \(-0.310799\pi\)
0.560005 + 0.828489i \(0.310799\pi\)
\(74\) 2.52559e8 0.979083
\(75\) 0 0
\(76\) −5.19988e8 −1.78786
\(77\) −4.31082e8 −1.39750
\(78\) 5.08331e8 1.55497
\(79\) 6.32088e8 1.82581 0.912905 − 0.408171i \(-0.133833\pi\)
0.912905 + 0.408171i \(0.133833\pi\)
\(80\) 0 0
\(81\) 4.30467e7 0.111111
\(82\) 1.41563e8 0.345771
\(83\) −3.09787e7 −0.0716493 −0.0358247 − 0.999358i \(-0.511406\pi\)
−0.0358247 + 0.999358i \(0.511406\pi\)
\(84\) 5.10139e8 1.11797
\(85\) 0 0
\(86\) 1.02020e9 2.01115
\(87\) −4.87633e8 −0.912549
\(88\) −1.24742e9 −2.21738
\(89\) −3.96023e7 −0.0669060 −0.0334530 − 0.999440i \(-0.510650\pi\)
−0.0334530 + 0.999440i \(0.510650\pi\)
\(90\) 0 0
\(91\) −1.03268e9 −1.57863
\(92\) −1.34215e9 −1.95324
\(93\) 6.39866e8 0.886984
\(94\) −4.05552e8 −0.535761
\(95\) 0 0
\(96\) −1.11094e8 −0.133497
\(97\) 9.30586e8 1.06729 0.533646 − 0.845708i \(-0.320821\pi\)
0.533646 + 0.845708i \(0.320821\pi\)
\(98\) −1.03475e7 −0.0113323
\(99\) 4.43767e8 0.464298
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 75.10.a.k.1.1 yes 4
3.2 odd 2 225.10.a.r.1.4 4
5.2 odd 4 75.10.b.h.49.2 8
5.3 odd 4 75.10.b.h.49.7 8
5.4 even 2 75.10.a.j.1.4 ✓ 4
15.2 even 4 225.10.b.n.199.7 8
15.8 even 4 225.10.b.n.199.2 8
15.14 odd 2 225.10.a.t.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
75.10.a.j.1.4 ✓ 4 5.4 even 2
75.10.a.k.1.1 yes 4 1.1 even 1 trivial
75.10.b.h.49.2 8 5.2 odd 4
75.10.b.h.49.7 8 5.3 odd 4
225.10.a.r.1.4 4 3.2 odd 2
225.10.a.t.1.1 4 15.14 odd 2
225.10.b.n.199.2 8 15.8 even 4
225.10.b.n.199.7 8 15.2 even 4